Is it (believed to be) possible, in the various standard examples of groups in which discrete log/Diffie Hellman are hard (including multiplicative groups in modular arithmetic and elliptic curves, and including cases in which Decisional Diffie Hellman is easy) to generate tuples of the form $(g, g^a, g^b, g^{ab})$ without in a sense "knowing" either a or b?

More formally, if there is a polynomial-time Turing machine $T(G, c)$ such that $T$ maps any input into such a tuple in the group described by $G$ ($G$ could just be a modulus if this is a modular multiplicative group, and $c$ could be thought of as random bits), must there exist a polynomial-time $T'$ such that $T'$, with the same input as $T$, outputs $(g, g^a, g^b, g^{ab}, a)$ or $(g, g^a, g^b, g^{ab}, b)$?

Clearly, the answer to this question is not known since there is always such a $T'$ if discrete log is easy and there is is a $T$ without such a $T'$ if Diffie Hellman is easy and discrete log is hard. I'm particularly interested in whether there is some existence result that says there must be a $T$ with no such $T'$ (under an assumption like hardness of discrete log), or whether there is a general conjecture that such a $T'$ always exists (or better yet, that this is implied by some other, widely believed, conjecture).

Cross-post from CSTheory stackexchange: https://cstheory.stackexchange.com/questions/8245/generating-a-diffie-hellman-tuple-without-being-able-to-know-one-of-the-discret